But I mean let's be honest here, are you really going to accept that integration worked? I'd like to believe it but experience tells me probably not. Because of the backfire effect I don't think you're likely to be convinced by some guy on the internet quoting social science research (especially since no research is perfect.) So here, let me give you a more human element. Here's three This American Life episodes, the first two are explicitly about integration and the third is about kids from a poor, public school visiting a rich, private school which is slightly less topical but helps to get the point across I think. Just listen and consider please.
https://www.thisamericanlife.org/radio-archives/episode/562/...
https://www.thisamericanlife.org/radio-archives/episode/563/...
https://www.thisamericanlife.org/radio-archives/episode/550/...
The fact that the gap is larger in segregated districts is difficult to interpret without knowing if the comparison is to the population as a whole (i.e. blacks in the district to whites as a whole) or within the district (i.e. blacks in the district to whites in the district). I'd also want to know if how those districts compared to other districts. Were these high achieving schools? Low achieving? Somewhere in the middle?